Understanding the Linearity Test for Inner Products

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Discussion Overview

The discussion revolves around the verification of a specific function as a valid inner product, focusing on the linearity test. Participants explore the implications of the function's structure and its adherence to the properties required for inner products.

Discussion Character

  • Technical explanation, Debate/contested, Mathematical reasoning

Main Points Raised

  • One participant presents a function defined as = x1x2 + y1y2 and questions its validity as an inner product based on the linearity test.
  • Another participant suggests explicitly calculating to check if it equals + , indicating a method to test linearity.
  • A participant raises a question about whether equals 0*, implying a consideration of scalar multiplication in the context of inner products.
  • Responses indicate a disagreement on whether the expression leads to a valid conclusion regarding linearity.
  • A later reply reveals a misunderstanding of the inner product's interpretation by one participant.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the validity of the inner product or the implications of the linearity test, with multiple competing views and interpretations present throughout the discussion.

Contextual Notes

Some assumptions about the definitions of inner products and linearity may be missing, and the discussion reflects varying interpretations of the mathematical expressions involved.

JamesGoh
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In one of my tutorial problems, I was asked to verify if the following function
is a valid inner product

<x,y>= x1x2 + y1y2

Note, x=(x1,x2)^{T} and y=(y1,y2)^{T}

where T means transpose of the matrix

The tutor said to us the answer is no because it fails the linearity test

Does it fail the linearity test because of the x1 and y1 terms in front of the x2 and y2 ?
 
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Try explicitly calculating
<br /> \langle x+y,z\rangle <br />
and see if it really does equal
<br /> \langle x,z\rangle +\langle y,z\rangle<br />
If it doesn't then you know it does not satisfy linearity.
 
Does <x,0*y>=0*<x,y>?
 
Yes, it does. But that doesn't prove anything.
 
HallsofIvy said:
Yes, it does.
No it doesn't. <x,0>=x1x2.
 
Oh, Blast! I was interpreting the given inner product wrong!
 

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